PHYSICAL REVIEW A 90, 022110 (2014)

Quantum regression theorem and non-Markovianity of quantum dynamics

Giacomo Guarnieri,1,2 Andrea Smirne,3,4 and Bassano Vacchini1,2

1Dipartimento di Fisica, Universita degli Studi di Milano, Via Celoria 16, 20133 Milan, Italy2Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Via Celoria 16, 20133 Milan, Italy

3Dipartimento di Fisica, Universita degli Studi di Trieste, Strada Costiera 11, 34151 Trieste, Italy4Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy

(Received 12 June 2014; published 15 August 2014)

We explore the connection between two recently introduced notions of non-Markovian quantum dynamicsand the validity of the so-called quantum regression theorem. While non-Markovianity of a quantum dynamicshas been defined looking at the behavior in time of the statistical operator, which determines the evolutionof mean values, the quantum regression theorem makes statements about the behavior of system correlationfunctions of order two and higher. The comparison relies on an estimate of the validity of the quantum regressionhypothesis, which can be obtained exactly evaluating two-point correlation functions. To this aim we considera qubit undergoing dephasing due to interaction with a bosonic bath, comparing the exact evaluation of thenon-Markovianity measures with the violation of the quantum regression theorem for a class of spectral densities.We further study a photonic dephasing model, recently exploited for the experimental measurement of non-Markovianity. It appears that while a non-Markovian dynamics according to either definition brings with itselfviolation of the regression hypothesis, even Markovian dynamics can lead to a failure of the regression relation.

DOI: 10.1103/PhysRevA.90.022110 PACS number(s): 03.65.Yz, 42.50.Lc, 03.67.−a

I. INTRODUCTION

In recent times there has been a revival in the studyof the characterization of non-Markovianity for an openquantum system dynamics. While the subject was naturallyborn together with the introduction of the first milestones inthe description of the time evolution of a quantum systeminteracting with an environment [1,2], the difficulty inherentin the treatment led to very few general results, and the verydefinition of a convenient notion of Markovian open quantumdynamics was not agreed upon. The focus initially was onfinding the closest quantum counterpart of the classical notionof Markovianity for a stochastic process, so that referencewas made to correlation functions of all order for the process.Recent work was rather focused on proposals of a notion ofMarkovian quantum dynamics based on an analysis of thebehavior of the statistical operator describing the system ofinterest only, thus concentrating on features of the dynamicalevolution map, which only determines mean values. Differentproperties of the time evolution map have been considered inthis respect [3–12]. In particular two viewpoints [4,6] appearto have captured important aspects in the characterization ofa dynamics which can be termed non-Markovian in the sensethat it relates to memory effects.

The aim of our work is to analyze the relationshipbetween these approaches and the validity of the so-calledquantum regression theorem [13,14], according to which thebehavior in time of higher order correlation functions can bepredicted building on the knowledge of the dynamics of themean values for a generic observable. The analysis can beperformed introducing a suitable quantifier for the violationof the quantum regression hypothesis, which in turn requiresknowledge of the exact two-time correlation functions. Wetherefore consider a two-level system coupled to a bosonicbath through a decoherence interaction, exactly estimatingfor a general class of spectral densities the predictions ofdifferent criteria for non-Markovianity of a dynamics and the

violation of the regression theorem. We further apply thisanalysis to a dephasing model, whose realization has beenrecently exploited to experimentally observe quantum non-Markovianity [15]. In both cases we show that the quantumregression theorem can be violated even in the presence ofa quantum dynamics which, according to either criteria, isconsidered Markovian.

The paper is organized as follows. In Sec. II we recall tworecently introduced notions of Markovianity for a quantumdynamics and the associated measures, while in Sec. III weaddress the formulation of the quantum regression theoremand introduce a simple estimator for its violation. We apply thisformalism to the pure dephasing spin boson model in Sec. IVdiscussing the relationship between the two approaches, andextend the analysis to a photonic dephasing model in Sec. V.We finally comment on our results in Sec. VI.

II. NON-MARKOVIANITY DEFINITIONS AND MEASURES

Let us start by briefly recalling the main features of thenotion of non-Markovian quantum dynamics which will beexploited in the following analysis. In the classical theory ofstochastic processes, the definition of Markov process involvesthe entire hierarchy of n-point joint probability distributionsassociated with the process. Since such a definition cannot bedirectly transposed to the quantum realm [16,17], differentand nonequivalent notions of quantum Markovianity havebeen introduced [3–11], along with different measures toquantify the degree of non-Markovianity of a given dynamics(see [18,19] for a very recent comparison). These definitionsall convey the idea that the occurrence of memory effects isthe proper attribute of non-Markovian dynamics, relying ondifferent properties of the dynamical maps which describe theevolution of the open quantum system. In the absence of initialcorrelations between the open system and its environment, i.e.,

ρSE(0) = ρS(0) ⊗ ρE(0) (1)

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GUARNIERI, SMIRNE, AND VACCHINI PHYSICAL REVIEW A 90, 022110 (2014)

with ρE(0) assumed to be fixed, the evolution of an openquantum system is characterized by a one-parameter familyof completely positive and trace preserving (CPT) maps{�(t)}t�0, such that [13]

ρS(t) = �(t)ρS, (2)

where ρS ≡ ρS(0) is the state of the open system at theinitial time t0 = 0. A relevant class of open quantum systemdynamics is provided by the semigroup ones, which arecharacterized by the composition law

�(t)�(s) = �(t + s) ∀t,s � 0. (3)

The generator of a semigroup of CPT maps is fixed bythe Gorini-Kossakowski-Sudarshan-Lindblad theorem [1,20],which implies that the dynamics of the system is given by theLindblad equation

d

dtρS(t) = −i[H,ρS(t)]

+∑

k

γk

(LkρS(t)L†

k − 1

2{L†

kLk,ρS(t)})

(4)

with γk � 0. The semigroups of CPT maps are identified withthe Markovian time-homogeneous dynamics according to allthe previously mentioned definitions of Markovianity, so thatthe differences between them actually concern the notion oftime-inhomogeneous Markovian dynamics.

In the following, we will take into account two definitionsof Markovianity and the corresponding measures of non-Markovianity. One definition [4] is related with the contrac-tivity of the trace distance under the action of the dynamicalmaps, while the other [6] relies on a divisibility property of thedynamical maps, which reduces to the semigroup compositionlaw in the time-homogeneous case.

A. Trace-distance measure

The basic idea behind the definition of non-Markovianityintroduced by Breuer, Laine, and Piilo (BLP) [4] is that achange in the distinguishability between the reduced statescan be read in terms of an information flow between the opensystem and the environment. The distinguishability betweenquantum states is quantified through the trace distance [21],which is the metric on the space of states induced by the tracenorm:

D(ρ1,ρ2) = 1

2‖ρ1 − ρ2‖1 = 1

2

∑k

|xk|, (5)

where the xk are the eigenvalues of the traceless Hermitianoperator ρ1 − ρ2. The trace distance takes values between0 and 1 and, most importantly, it is a contraction under theaction of CPT maps. By investigating the evolution of thetrace distance between two states of the open system coupledto the same environment but evolved from different initialconditions,

D(t,ρ

1,2S

) ≡ D(ρ1

S(t),ρ2S(t)

), ρk

S(t) = �(t)ρkS, (6)

one can thus describe the exchange of information betweenthe open system and the environment. A decrease of thetrace distance D(t,ρ1,2

S ) means a lower ability to discriminate

between the two initial conditions ρ1S and ρ2

S , which can beexpressed by saying that some information has flown out ofthe open system. On the same ground, an increase of thetrace distance can be ascribed to a backflow of informationto the open system and then represents a memory effect inits evolution. Non-Markovian quantum dynamics can be thusdefined as those dynamics which present a nonmonotonicbehavior of the trace distance, i.e., such that there are timeintervals �+ in which

σ(t,ρ

1,2S

) = d

dtD

(t,ρ

1,2S

)> 0. (7)

Consequently, the non-Markovianity of an open quantumsystem’s dynamics {�(t)}t�0 is quantified by the measure

N = maxρ

1,2S

∫�+

σ(t,ρ

1,2S

)dt. (8)

The maximization involved in the definition of this measurecan be greatly simplified since the optimal states must beorthogonal [22] and, even more, one can determine N bymeans of a local maximization over one state only [23].This measure of non-Markovianity has been also investigatedexperimentally in all-optical settings [15,24,25].

B. Divisibility measure

The definition given by Rivas, Huelga, and Plenio (RHP) [6]identifies Markovian dynamics with those dynamics whichare described by a CP-divisible family of quantum dynamicalmaps {�(t)}t�0 (CP standing for completely positive), i.e.,such that

�(t2) = �(t2,t1)�(t1) ∀ t2 � t1 � 0, (9)

�(t2,t1) being itself a completely positive map. Indeed,if �(t2,t1) = �(t2 − t1) the composition law in Eq. (9) isequivalent to the semigroup composition law. An importantproperty of this definition is that, provided that the evolutionof the reduced state can be formulated by a time-local masterequation

d

dtρS(t) = K(t)[ρS(t)] = −i[H (t),ρS(t)]

+∑

k

γk(t)(Lk(t)ρS(t)L†

k(t)

− 1

2{L†

k(t)Lk(t),ρS(t)})

, (10)

the positivity of the coefficients, γk(t) � 0 for any t � 0,is equivalent to the CP divisibility of the correspondingdynamics. This can be shown by taking into account the familyof propagators �(t2,t1) associated with Eq. (10),

�(t2,t1) = T← exp

(∫ t2

t1

K(s)ds

), (11)

where T← denotes the time ordering and �(t,0) ≡ �(t). Byconstruction, the propagators �(t2,t1) satisfy Eq. (9), but, ingeneral, they are not CP maps. One can show [26,27] that thepropagators are actually CP if and only if the coefficients γk(t)are positive functions of time.

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QUANTUM REGRESSION THEOREM AND NON- . . . PHYSICAL REVIEW A 90, 022110 (2014)

The corresponding measure of non-Markovianity is givenby

I =∫R+

dt g(t) (12)

with

g(t) = limε→0+

1N

‖�Choi(t,t + ε)‖1 − 1

ε, (13)

where �Choi is the Choi matrix associated with �. Given amaximally entangled state between the system and an ancilla,|ψ〉 = 1√

N

∑Nk=1 |uk〉 ⊗ |uk〉, one has [28]

�Choi = N (� ⊗ 1N )(|ψ〉〈ψ |). (14)

The positivity of the Choi matrix corresponds to the completepositivity of the map � and it is equivalent to the condition‖�Choi‖1 = N , so that the quantity g(t) is different from zeroif and only if the CP divisibility of the dynamics is broken.

Finally, since the trace distance is contractive under CPTmaps, if a dynamics is Markovian according to the RHPdefinition, then it is so also according to the BLP definition,i.e.,

I = 0 =⇒ N = 0, (15)

while the opposite implication does not hold [26,29,30].

III. THE QUANTUM REGRESSION THEOREM

As recalled in the Introduction, the quantum regressiontheorem provides a benchmark structure in order to study themultitime correlation functions of an open quantum system.For the sake of simplicity, we focus on the two-time correlationfunctions only. Given two open system operators, A ⊗ 1E andB ⊗ 1E , where 1E denotes the identity on the Hilbert spaceassociated with the environment, their two-time correlationfunction is defined as

〈A(t2)B(t1)〉 ≡ Tr[U †(t2)A ⊗ 1EU (t2)

×U †(t1)(B ⊗ 1E)U (t1)ρSE(0)], (16)

where U (t) is the overall unitary evolution operator and weset t2 � t1 � 0. In the following, we assume an initial stateas in Eq. (1), as well as a time-independent total HamiltonianHT = HS ⊗ 1E + 1S ⊗ HE + HI , so that U (t) = e−iHT t .

The condition of an initial product state with a fixedenvironmental state guarantees the existence of a reduceddynamics; see Eqs. (1) and (2). This means that all theone-time probabilities associated with the observables of theopen systems and, as a consequence, their mean values can beevaluated by means of the family of reduced dynamical mapsonly, without need for any further reference to the overallunitary dynamics. An analogous result holds for the two-timecorrelation functions, if one can apply the so-called quantumregression theorem. The latter essentially states that underproper conditions the dynamics of the two-time correlationfunctions can be reconstructed from the dynamics of the meanvalues, or, equivalently, of the statistical operator. Indeed, ifthe quantum regression theorem cannot be applied, one needsto come back to the full unitary dynamics in order to determinethe evolution of the two-time correlation functions. We will not

repeat here the detailed derivation of the quantum regressiontheorem, which can be found in [13,14,31]. Nevertheless, letus recall the basic ideas. First, by introducing the operator

χ (t2,t1) = e−iHT (t2−t1)B ⊗ 1EρSE(t1)eiHT (t2−t1), (17)

the two-time correlation function in Eq. (16) can be rewrittenas

〈A(t2)B(t1)〉 = TrS A TrE χ (t2,t1). (18)

Now, suppose that we can describe the evolution of χ (t2,t1)with respect to t2 with the same dynamical maps which fix theevolution of the statistical operator, i.e.,

χ (t2,t1) = �(t2,t1)[χ (t1,t1)], (19)

where �(t2,t1) is the propagator introduced in Eq. (11). Then,Eq. (18) directly provides

〈A(t2)B(t1)〉qrt = TrS A�(t2,t1)[BρS(t1)]. (20)

The two-time correlation functions can be fully determined bythe dynamical maps which fix the evolution of the statisticaloperator: the validity of Eq. (20) can be identified with thevalidity of the quantum regression theorem and we will usethe subscript qrt to denote the two-time correlation functionsevaluated through Eq. (20). Indeed, all the procedure relies onEq. (19), which requires that the same assumptions made inorder to derive the dynamics of ρS(t) can be made also to getthe evolution of χ (t2,t1) with respect to t2 [14]. Especially, thehypothesis of an initial total product state in Eq. (1) turns intothe hypothesis of a product state at any intermediate time t1,

ρSE(t1) = ρS(t1) ⊗ ρE. (21)

The physical idea is that the quantum regression theoremholds when the system-environment correlations due to theinteraction can be neglected [32]. Note that this conditionwill never be strictly satisfied, as long as the system and theenvironment mutually interact, but it should be understood asa guideline to detect the regimes in which Eq. (20) provides asatisfying description of the evolution of the two-time correla-tion functions. More precisely, Dumcke [33] demonstrated thatthe exact expression of the two-time (multitime) correlationfunctions, see Eq. (16), converges to the expression in Eq. (20)in the weak coupling limit and in the singular coupling limit. Asis well known, in these limits the reduced dynamics convergesto a semigroup dynamics [34,35]. Nevertheless, the correctnessof a semigroup description of the reduced dynamics is notalways enough to guarantee the accuracy of the quantumregression theorem [36,37]. More in general, the precise linkbetween a sharply defined notion of Markovianity of quantumdynamics and the quantum regression theorem has still to beinvestigated.

The quantum regression theorem provided by Eq. (20)can be equivalently formulated in terms of the differentialequations satisfied by mean values and two-time correlationfunctions, as was originally done in [38]. For the sake ofsimplicity, let us restrict ourselves to the finite-dimensionalcase, i.e., the Hilbert space associated with the open system isCN . Consider a reduced dynamics fixed by the family of maps{�(t)}t�0 and a basis {Ei}1,...,N2 of linear operators on CN ,such that the corresponding mean values fulfill the coupled

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GUARNIERI, SMIRNE, AND VACCHINI PHYSICAL REVIEW A 90, 022110 (2014)

linear equations of motion [31]

d

dt〈Ei(t)〉 =

∑j

Gij (t)〈Ej (t)〉, (22)

with the initial condition 〈Ei(t)〉|t=0 = 〈Ei(0)〉. In this case,the quantum regression theorem is said to hold if the two-timecorrelation functions satisfy [13,14]

d

dt2〈Ei(t2)Ek(t1)〉qrt =

∑j

Gij (t2)〈Ej (t2)Ek(t1)〉qrt , (23)

with the initial condition

〈Ei(t2)Ek(t1)〉|t2=t1 = 〈Ei(t1)Ek(t1)〉.In the following, we will compare the evolution of the exact

two-time correlation functions obtained from the full unitaryevolution 〈Ei(t2)Ek(t1)〉, see Eq. (16), with those predicted bythe quantum regression theorem 〈Ei(t2)Ek(t1)〉qrt . To quantifythe error made by using the latter, we exploit the relative error;i.e., we use the following figure of merit:

Z ≡∣∣∣∣1 − 〈A(t2)B(t1)〉qrt

〈A(t2)B(t1)〉∣∣∣∣, (24)

which depends on the chosen couple of open system operators.Hence, in general, one should consider different estimators,one for each couple of operators in the basis {Ei}1,...,N2 , anda maximization over them could be taken. Nevertheless, inthe following analysis it will be enough to deal with a singlecouple of system operators, which fully encloses the violationsof the quantum regression theorem for the models at hand.

IV. PURE-DEPHASING SPIN BOSON MODEL

In this section, we take into account a model whose fullunitary evolution can be exactly evaluated [13,39], so as to ob-tain the exact expression of the two-time correlation functions,to be compared with the expression provided by the quantumregression theorem. This model is a pure-decoherence model,in which the decay of the coherences occurs without a decayof the corresponding populations. Indeed, this is due to thefact that the free Hamiltonian of the open system HS ⊗ 1E

commutes with the total Hamiltonian HT [13].

A. The model

Let us consider a two-level system linearly interacting witha bath of harmonic oscillators, so that the total Hamiltonian is

HT = ωs

2σz ⊗ 1E + 1S ⊗

∑k

ωkb†kbk

+∑

k

σz ⊗ (gkb†k + g∗

k bk). (25)

The unitary evolution operator of the overall system in theinteraction picture is given by [13]

U (t) = ei�(t)V (t), (26)

where the first factor is an irrelevant global phase and thesecond factor is the unitary operator

V (t) = exp

[1

2σz ⊗

∑k

(αk(t)b†k − α∗k (t)bk)

], (27)

with

αk(t) = 2gk

1 − eiωkt

ωk

. (28)

The reduced dynamics is readily calculated to give

ρS(t) =(

ρ00 ρ01γ (t)e−iωs t

ρ10γ∗(t)eiωs t ρ11

), (29)

where the function γ (t) is given by

γ (t) = TrE ρE

∏k

exp[αk(t)b†k − α∗k (t)bk]

= TrE ρE

∏k

(αk(t)), (30)

(α) being the displacement operator of argument α [40]. Theassociated master equation reads

d

dtρS(t) = −i

ε(t)

2[σz,ρS(t)] + D(t)

2[σzρS(t)σz − ρS(t)],

(31)

where

ε(t) = ωs − Im

[dγ (t)/dt

γ (t)

](32)

and the so-called dephasing function D(t) is

D(t) = −Re

[dγ (t)/dt

γ (t)

]= − d

dtln |γ (t)|. (33)

In the following, we will focus on the case of an initialthermal state of the bath, ρE = exp(−βHE)/Z with Z =TrE exp(−βHE) and β = (kBT )−1 the inverse temperature.We also consider the continuum limit: given a frequencydistribution f (ω) of the bath modes, we introduce the spectraldensity J (ω) = 4f (ω)|g(ω)|2, so that one has [13]

γ (t) = exp

[−

∫ ∞

0dω J (ω) coth

(βω

2

)1 − cos(ωt)

ω2

],

(34)and hence ε(t) = ωs and

D(t) =∫ ∞

0dω J (ω) coth

(βω

2

)sin(ωt)

ω. (35)

B. Measures of non-Markovianity

1. General expressions

For this specific model, the two definitions of Markovianityare actually equivalent [41]; i.e., not only Eq. (15) holds, butalso the opposite does so. This is due to the fact that thereis only one operator contribution in the time-local masterequation (31), corresponding to the dephasing interaction.Nevertheless, the numerical values of the two measures of non-Markovianity are in general different and, more importantly,they depend in a different way on the parameters of the model.

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Let us start by evaluating the BLP measure; see Sec. II A.The trace distance between two reduced states evolved throughEq. (29) is given by

D(t,ρ

1,2S

) =√

δ2p + |δc|2|γ (t)|2, (36)

where δp = ρ100 − ρ2

00 and δc = ρ101 − ρ2

01 are the differencesbetween, respectively, the populations and the coherences ofthe two initial conditions ρ1

S and ρ2S . The couple of initial

states that maximizes the growth of the trace distance is givenby the pure orthogonal states ρ

1,2S = |ψ±〉〈ψ±|, where |ψ±〉 =

1√2(|0〉 ± |1〉), and the corresponding trace distance at time t

is simply |γ (t)|. The BLP measure therefore reads

N =∑m

[|γ (bm)| − |γ (am)|], (37)

where �+ = ⋃m(am,bm) is the union of the time intervals in

which |γ (t)| increases. The BLP measure is different from zeroif and only if d|γ (t)|/dt > 0 for some interval of time, which isequivalent to the requirement that the dephasing function D(t)in Eq. (31) is not a positive function of time, i.e., that the CPdivisibility of the dynamics is broken; Sec. II B. As anticipated,for this model N > 0 ⇐⇒ I > 0. Furthermore, given a puredephasing master equation as in Eq. (31), one has [6] g(t) = 0ifD(t) � 0 and g(t) = −D(t) ifD(t) < 0, so that, see Eq. (33),

I =∑m

[ln |γ (bm)| − ln |γ (am)|], (38)

where the am and bm are defined as for the BLP measure.

2. Zero-temperature environment

In order to evaluate explicitly the non-Markovianity mea-sures, we need to specify the spectral density J (ω). In thefollowing, we assume a spectral density of the form

J (ω) = λωs

�s−1e− ω

� , (39)

where λ is the coupling strength, the parameter s fixes thelow-frequency behavior, and � is a cutoff frequency. The non-Markovianity for the pure dephasing spin model with a spectraldensity as in Eq. (39) has been considered in [18,42] for thecase λ = 1. We are now interested in the comparison betweennon-Markovianity and violations of the quantum regressiontheorem, so that, as will become clear in the next section, thedependence on λ plays a crucial role. In particular, we considerthe case of low temperature, i.e., β � 1, so that coth( βω

2 ) ≈ 1.The dephasing function in this case reads, see Eq. (35),

Ds(t) = �(s)

[1 + (�t)2]s2

sin[s arctan(�t)], (40)

with �(s) the Euler gamma function, which can be expressedin the equivalent but more compact form, see the Appendix,

Ds(t) = �(s)Im[(1 + i�t)s]

[1 + (�t)2]s. (41)

Correspondingly, the decoherence function can be written as

γs(t) = exp

[− λ�(s − 1)

(1 − Re[(1 + i�t)s−1]

[1 + (�t)2]s−1

)]. (42)

As before, let �+ be the union of the time intervals for whichD(t) < 0, i.e., equivalently, |γ (t)| increases. The number ofsolutions of the equation D(t) = 0 grows with the parameter s:for s = 1,2 the dephasing function is always strictly positive,while for s = 3 and s = 4 there is one zero at t∗3 =

√3

�and

t∗4 = 1�

, respectively. Indeed, if the number of zeros is odd,D(t) is negative from its last zero to infinity, while if the numberof zeros is even, it approaches zero asymptotically from above.As a consequence, the two measures of non-Markovianity areequal to zero for s = 1,2 and, to give an example, one has fors = 3

N3(λ) = limt→∞ |γ (t)| − |γ (t∗3 )| = e−λ − e− 9

8 λ,

(43)I3(λ) = lim

t→∞ ln |γ (t)| − ln |γ (t∗3 )| = λ

8,

and, analogously, for s = 4

N4(λ) = e−2λ − e− 52 λ, I4(λ) = λ

2. (44)

In Figs. 1(a) and 1(b), we report, respectively, the BLP andthe RHP measures of non-Markovianity as a function of λ, fordifferent values of s.

The behavior of the two measures is clearly different. TheRHP measure is a monotonically increasing function of both

FIG. 1. (Color online) (a) BLP measure of non-MarkovianityNs(λ), see Eq. (37), and (b) RHP measure of non-MarkovianityIs(λ), see Eq. (38), as a function of the coupling strength λ forincreasing values of the parameter s. In both panels the curves areevaluated for s = 3 (black thick solid line), s = 3.5 (blue solid line),s = 4 (magenta dashed line), s = 4.5 (green dashed thick line), s = 5(red dot-dashed line), and s = 5.5 (orange dotted line).

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FIG. 2. (Color online) (a) BLP measure of non-MarkovianityNs(λ), see Eq. (37), as a function of the parameter s, for λ = 1.(b) and (c) Decoherence function γs(t) as a function of rescaled forλ = 0.5 and different values of s (b), and for s = 4 and differentvalues of λ (c).

λ and s: the increase is linear with respect to the formerparameter and exponential with respect to the latter. On theother hand, for every fixed s, there is a critical value ofthe coupling strength λ∗(s), which is smaller for increasings, that separates two different regimes of the BLP measure:for λ < λ∗(s), the non-Markovianity measure increases withthe increase of the system-environment coupling, while forλ > λ∗(s) it decreases with the increase of the coupling.Analogously, there is a threshold value s∗(λ) of the parameters, which is higher for smaller values of λ, such that the BLPmeasure increases for s < s∗(λ) and decreases for s > s∗(λ);see also Fig. 2(a). Incidentally, the maximum value as afunction of λ, maxλ Ns(λ), is a monotonically increasingfunction of the parameter s. Indeed, the different behavior ofthe non-Markovianity measures traces back to their differentfunctional dependence of the decoherence function γs(t),which is plotted in Figs. 2(b) and 2(c) for different valuesof s and λ. One can see how γs(t) takes on smaller valueswithin [0,1] for growing values of λ, while its global minimum

decreases with increasing s. Now, while the BLP measure isfixed by the difference between the values of γs(t) at the edgesof the time intervals [am,bm] in which γs(t) increases, seeEq. (37), the RHP measure is fixed by the ratio between thesame values, see Eq. (38). Hence, as the coupling strengthgrows over the threshold λ∗(s) or the parameter s overcomesthe threshold s∗(λ), the difference between bm and am isincreasingly smaller, and therefore Ns(λ) is so. However,the ratio between bm and am always increases with λ ands, as witnessed by the corresponding monotonic increaseof Is(λ).

C. Validity of regression hypothesis

1. Exact expression versus quantum regression theorem

The exact unitary evolution, Eq. (26), directly provides uswith the average values, as well as the two-time correlationfunctions of the observables of the system. In view ofthe comparison with the description given by the quantumregression theorem, see Sec. III, let us focus on the basis oflinear operators onC2, orthonormal with respect to the Hilbert-Schmidt scalar product, given by {1/√2,σ−,σ+,σz/

√2}.

Indeed, the first and the last elements of the basis are constantof motion, see Eq. (29), while the mean values of σ− and σ+evolve according to, respectively,

〈σ−(t)〉 = γ (t)e−iωs t 〈σ−(0)〉 (45)

and the complex conjugate relation. In a similar way, allthe two-time correlation functions involving 1/

√2 or σz/

√2

satisfy the condition of the quantum regression theorem ina trivial way, as at most one operator within the two-timecorrelation function actually evolves. The only nontrivialexpressions are thus the following:

〈σ−(t2)σ+(t1)〉 = e−iωs (t2−t1)γ (t2,t1)eiφ(t2,t1)〈(σ−σ+)(t1)〉,(46)

〈σ+(t2)σ−(t1)〉 = eiωs (t2−t1)γ ∗(t2,t1)eiφ(t2,t1)〈(σ+σ−)(t1)〉,where

γ (t2,t1) = TrE ρE

∏k

(αk(t2) − αk(t1)) (47)

and

φ(t2,t1) =∑

k

Im[α∗k (t2)αk(t1)]. (48)

Here, to derive (46) we used the properties of the displacementoperator [40]

(α) (β) = (α + β)eiIm(αβ∗), †(α) = (−α),

and the equality 〈(σ+σ−)(t)〉 = 〈σ+σ−〉.We can now obtain the corresponding two-time correlation

functions as predicted by the quantum regression theorem. ByEq. (45), one has

d

dt〈σ−(t)〉 =

(dγ (t)/dt

γ (t)− iωs

)〈σ−(t)〉 (49)

and the complex conjugate relation for 〈σ+(t)〉. The specificchoice of the operator basis has lead us to a diagonal matrix G

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QUANTUM REGRESSION THEOREM AND NON- . . . PHYSICAL REVIEW A 90, 022110 (2014)

in Eq. (22). Hence, one has immediately

〈σ−(t2)σ+(t1)〉qrt = e−iωs (t2−t1) γ (t2)

γ (t1)〈σ−(t1)σ+(t1)〉,

(50)

〈σ+(t2)σ−(t1)〉qrt = eiωs (t2−t1) γ∗(t2)

γ ∗(t1)〈σ+(t1)σ−(t1)〉.

The quantum regression theorem will be generally violatedwithin this model; compare Eq. (46) and (50). We quantifysuch a violation by means of the figure of merit introduced inEq. (24), which for the couple of operators σ− and σ+ reads

Z =∣∣∣∣1 − 〈σ−(t2)σ+(t1)〉qrt

〈σ−(t2)σ+(t1)〉∣∣∣∣

=∣∣∣∣1 − γ (t2)

γ (t1)γ (t2,t1)eiφ(t2,t1)

∣∣∣∣. (51)

2. Quantitative analysis of the violations of the quantumregression theorem

The expressions of the previous paragraph hold for genericinitial state of the bath and spectral density. Now, we come backto the specific choice of an initial thermal bath. The results inEq. (50) are in this case in agreement with those found in [43],where the two-time correlation functions have been evaluatedfocusing on a spectral density as in Eq. (39) with s = 1, whilekeeping a generic temperature of the bath. Instead, we willfocus on the case T = 0 and maintain a generic value of s

in order to compare the behavior of the two-time correlationfunctions with the measures of non-Markovianity.

First, note that by using the definition of the displacementoperator as well as Eq. (28), one can show the generalidentity

(αk(t2) − αk(t1)) = (αk(t2 − t1)eiωkt1 ). (52)

But then, since for a thermal state TrE (α)ρE is a function of|α| only [13], Eq. (52) implies

γ (t2,t1) = γ (t2 − t1); (53)

see Eqs. (47) and (30). In addition we have in the continuumlimit, see Eq. (48),

φ(t2,t1) =∫

dωJ (ω)

ω2{sin(ωt2) − sin(ωt1) − sin[ω(t2 − t1)]},

so that, for J (ω) as in Eq. (39) and using Eq. (35) in thezero-temperature limit, we get

φs(t2,t1) = [Ds−1(t2) − Ds−1(t1) − Ds−1(t2 − t1)]/�. (54)

The identities in Eqs. (41) and (42), along with Eqs. (53)and (54), finally provide us with the explicit expression of theestimator for the violations of the quantum regression theorem,see Eq. (51),

Zs(λ) = |1 − exp (λ�(s − 1){1 − [1 + i�(t2 − t1)]1−s

− (1 + i�t1)1−s + (1 + i�t2)1−s})|, (55)

whose behavior as a function of λ and s is shown in Figs. 3(a)and 3(b). The violation of the quantum regression theoremmonotonically increases with increasing values of both thecoupling strength λ and the parameter s. This behavior is

FIG. 3. (Color online) (a) Zs(λ) as a function of the parameter s

and of the coupling strength λ, see Eq. (55), for �t1 = 1 and �t2 = 2.(b) Section of (a) for s = 2,3,4.

clearly in agreement with that of the RHP measure of non-Markovianity; see Sec. IV B 2 and in particular Fig. 1. From aquantitative point of view there is, however, some difference asthe estimator Zs(λ), at variance with the RHP measure, growslinearly with λ only for small values of s, while it growthsfaster for s > 3; compare with Fig. 1(b). In any case, the RHPmeasure appears to be more directly related with the strength ofthe violation to the quantum regression theorem, as comparedwith the BLP measure. This can be traced back to the differentinfluence of the system-environment correlations on the twomeasures. As we recalled in Sec. III, the hypothesis that thestate of the total system at any time t is well approximatedby the product state between the state of the open systemand the initial state of the environment, see Eq. (21), lies atthe basis of the quantum regression theorem. This hypothesisis expected to hold in the weak coupling regime, while foran increasing value of λ, the interaction will build strongersystem-environment correlations, leading to a strong violationof the quantum regression theorem. The establishment ofcorrelations between the system and the environment due tothe interaction plays a significant role also in the subsequentpresence of memory effects in the dynamics of the opensystem [44–46]. Indeed, different signatures of the memoryeffects can be affected by system-environment correlationsin different ways. In particular, the CP divisibility of thedynamical maps appears to be a more fragile property thanthe contractivity of the trace distance and therefore it is moresensitive to the violations of the quantum regression theorem.Furthermore, it is worth noting that the estimator Zs(λ) steadilyincreases with the coupling strength λ even for values of s

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such that the corresponding reduced dynamics is Markovianaccording to either definitions. The validity of the quantumregression theorem calls therefore for stricter conditions thanthe Markovianity of quantum dynamics. In the case of thetrace distance criterion for non-Markovianity, which is directlybased on an observable quantity such as the distance in timebetween different system states, it is also important to stressthe different dependence on λ in quantifying non-Markovianityor violation of the quantum regression theorem, respectively.The monotonic dependence on the coupling strength in Zs(λ)is indeed not reflected in N (λ).

V. PHOTONIC REALIZATION OF DEPHASINGINTERACTION

In the pure dephasing spin-boson model, there is noregime in which the quantum regression theorem is strictlysatisfied, apart from the trivial case λ = 0. In addition, we haveshown that the strength of the violations of this theorem hasthe same qualitative behavior of the RHP non-Markovianitymeasure, as they increase with both λ and the parameters. In this section, we take into account a different puredephasing model, which allows us to deepen our analysison the relationship between the quantum regression theoremand the Markovianity of the reduced-system dynamics. Inparticular, we show that in general these two notions should beconsidered as different since the quantum regression theoremmay be strongly violated, even if the open system’s dynamicsis Markovian, irrespective of the exploited definition.

A. The model

Let us deal with the pure-dephasing interaction consideredin Ref. [15]. The open system here is represented by thepolarization degrees of freedom of a photon generated by spon-taneous parametric down conversion, while the environmentconsists in the corresponding frequency degrees of freedom.The overall unitary evolution, which is realized via a quartzplate that couples the polarization and frequency degrees offreedom, can be described as

U (t)|j,ω〉 = einj ωt |j,ω〉 j = 0,1, (56)

where |0〉 ≡ |H 〉 and |1〉 ≡ |V 〉 are the two polarization states(horizontal and vertical), with refractive indexes, respectively,n0 ≡ nH and n1 ≡ nV , while |ω〉 is the environmental statewith frequency ω. If we consider an initial product state,see Eq. (1), with a pure environmental state ρE = |�E〉〈�E |,where

|�E〉 =∫

dω f (ω)|ω〉, (57)

we readily obtain that the reduced dynamics is given byEq. (29). Again, we are in the presence of a pure dephasingdynamics, the only difference being the decoherence function,which now reads

γ (t) =∫

dω |f (ω)|2ei nωt , (58)

with n ≡ n1 − n0. For the rest, the results of Secs. IV Aand IV B directly apply also to this model: the master equationis given by Eq. (31), with ε(t) and D(t) as in, respectively,

Eq. (32) (for ωs = 0) and Eq. (33), while the non-Markovianitymeasures are as in Eq. (37) and Eq. (38). Analogously, thetwo-time correlation functions are given by Eq. (46) with

γ (t2,t1) = γ (t2 − t1) φ(t2,t1) = 0, (59)

while the application of the quantum regression theorem leadsto the expressions in Eq. (50) (with ωs = 0). Hence, theviolations of the quantum regression theorem can be quantifiedby

Z =∣∣∣∣1 − 〈σ−(t2)σ+(t1)〉qrt

〈σ−(t2)σ+(t1)〉∣∣∣∣ =

∣∣∣∣1 − γ (t2)

γ (t1)γ (t2 − t1)

∣∣∣∣. (60)

B. Lorentzian frequency distributions

1. Semigroup dynamics

Despite its great simplicity, this model allows us todescribe the transition between Markovian and non-Markoviandynamics in concrete experimental settings [15,24]. Differentdynamics are obtained for different choices of the initialenvironmental state, see Eq. (1) and the related discussion,i.e., for different initial frequency distributions, see Eq. (57).The latter can be experimentally set, e.g., by properly rotatinga Fabry-Perot cavity, through which a beam of photons gener-ated by spontaneous parametric down conversion passes [15].A natural benchmark is represented by the Lorentzian distri-bution

|f (ω)|2 = δω

π [(ω − ω0)2 + (δω)2], (61)

where δω is the width of the distribution and ω0 its central fre-quency, as this provides a reduced semigroup dynamics [46].The decoherence function, which is given by the Fouriertransform of the frequency distribution, see Eq. (58), is infact

γ (t) = e− n(δω−iω0)t . (62)

Thus, replacing this expression in Eqs. (32) and (33),one obtains a Lindblad equation, given by Eq. (31) withε(t) = − nω0 and D(t) = n δω. In addition, γ (t2 − t1) =γ (t2)/γ (t1) and hence, as one can immediately see by Eq. (60),Z = 0. For this model, as long as the reduced dynamics isdetermined by a completely positive semigroup, the quantumregression theorem is strictly valid. Let us emphasize that thisis the case even if the total state is not a product state at anytime t . For example if the initial state of the open system isthe pure state |ψS〉 = α|H 〉 + β|V 〉, with |α|2 + |β|2 = 1, thetotal state at time t is

|ψSE(t)〉 =∫

dωf (ω)(αeinH ωt |H,ω〉 + βeinV ωt |V,ω〉). (63)

This is an entangled state, of course unless α = 0 or β = 0;nevertheless, the quantum regression theorem does hold. Thisclearly shows that for the quantum regression theorem, asfor the semigroup description of the dynamics [45–47], theapproximation encoded in Eq. (21) should be considered asan effective description of the total state, which can be verydifferent from its actual form, even when the theorem is valid.

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QUANTUM REGRESSION THEOREM AND NON- . . . PHYSICAL REVIEW A 90, 022110 (2014)

2. Time-inhomogeneous Markovian and non-Markoviandynamics

Now, we consider a more general class of frequencydistributions; namely, the linear combination of two Lorentziandistributions,

|f (ω)|2 =∑j=1,2

Ajδωj

π [(ω − ω0,j )2 + (δωj )2], (64)

with A1 + A2 = 1. The decoherence function (58) is in thiscase

γ (t) = e− n(δω1−iω0,1)t + re− n(δω2−iω0,2)t

1 + r, (65)

with r ≡ A2A1

, while the estimator of the violations of thequantum regression theorem, see Eq. (60), can be written asa function of the difference between the central frequencies, ω = ω0,1 − ω0,2, as well as of the difference between thecorresponding widths, δω = δω1 − δω2. If we assume thatthe two central frequencies are equal, ω0,1 = ω0,2 = ω0, theevolution of the two-level statistical operator is fixed by atime-local master equation as in Eq. (31), with ε(t) = − nω0

and

D(t) = nδω1e

− nδω1t + r δω2e− nδω2t

e− nδω1t + r e− nδω2t. (66)

The latter is a positive function of time: the reduced dynamicsis CP divisible, see Sec. II B, and hence it is Markovianwith respect to both the BLP and RHP definitions. Indeed,now we are in the presence of a time-inhomogeneous Marko-vian dynamics. Nevertheless, as γ (t2 − t1) �= γ (t2)/γ (t1) thequantum regression theorem is violated; see Eq. (60). This isexplicitly shown in Fig. 4(a), where Z is plotted as a function of δω = δω1 − δω2 and nτ , with τ = t2 − t1. With growingdifference between the two widths, as well as the length ofthe time interval, the deviations from the quantum regressiontheorem are increasingly strong, up to a saturation value ofthe estimator Z. Contrary to the semigroup case, here, even ifthe dynamics is Markovian according to both definitions, theactual behavior of the two-time correlation functions cannotbe reconstructed by the evolution of the mean values.

Finally, let us consider a frequency distribution as inEq. (64), but now with δω1 = δω2 = δω and ω0,1 �= ω0,2. Thisfrequency distribution has two peaks and the resulting reduceddynamics is non-Markovian [15,46]. In this case the BLPnon-Markovianity measure (8) increases with the increasingof the distance between the two peaks, while the estimator Z

grows for small values of the distance and then it exhibits anoscillating behavior, see Fig. 4(b). Indeed, for ω = 0 onerecovers the semigroup dynamics previously described and,accordingly, Z goes to zero. Summarizing, by varying thedistance between the two peaks, one obtains a transition froma Markovian (semigroup) dynamics to a non-Markovian oneand, correspondingly, the quantum regression theorem ceasesto be satisfied and is even strongly violated. Nevertheless,the qualitative behavior of, respectively, the non-Markovianityof the reduced dynamics and the violation of the quantumregression theorem appear to be different.

FIG. 4. (Color online) Violation of the quantum regression the-orem, as quantified by the estimator Z in Eq. (60) (a) in the time-inhomogeneous Markovian case, ω0,1 = ω0,2 = ω0, as a function of δω = δω1 − δω2 and ω0τ = ω0(t2 − t1), for ω0t1 = 1 and r = 1;(b) in the non-Markovian case, δω1 = δω2 = δω, as a function of ω0 = ω0,1 − ω0,2 and δω τ , for δω t1 = 1 and r = 2. In both panels n = 1.

VI. CONCLUSIONS

We have explored the relationship between two criteriafor Markovianity of a quantum dynamics, namely the CPdivisibility of the quantum dynamical map and the behaviorin time of the trace distance between two distinct initialstates, and the validity of the quantum regression theorem,which is a statement relating the behavior in time of themean values and of the two-time correlation functions ofsystem operators. The first open system considered is atwo-level system affected by a bosonic environment througha dephasing interaction. For a class of spectral densities withexponential cutoff and power-law behavior at low frequencieswe have studied the onset of non-Markovianity as a functionof the coupling strength and of the power determining thelow-frequency behavior, further giving an exact expression forthe corresponding non-Markovianity measures. The deviationfrom the quantum regression theorem has been estimatedevaluating the relative error made in replacing the exacttwo-time correlation function for the system operators with theexpression reconstructed by the evolution of the correspondingmean values. It appears that the validity of the quantumregression theorem represents a stronger requirement than

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GUARNIERI, SMIRNE, AND VACCHINI PHYSICAL REVIEW A 90, 022110 (2014)

Markovianity, according to either criteria, which in this casecoincide but quantify non-Markovianity in a different way andexhibit distinct performances in their dependence on strengthof the coupling and low-frequency behavior. We have furtherconsidered an all-optical realization of a dephasing interaction,as recently exploited for the experimental investigation ofnon-Markovianity, obtaining also in this case, for differentchoices of the frequency distribution, significant violationsto the quantum regression theorem even in the presence ofa Markovian dynamics. This can be understood in terms ofthe different relevance of the quantum correlations betweensystem and environment in the development of the jointdynamics. While these correlations may not be strong enoughto induce a non-Markovian time development of the reducedstatistical operator, obtained by directly taking the partialtrace, they can still importantly affect the time development ofthe correlation functions, in which the partial trace is takenonly after considering the product of different Heisenbergoperators.

These results suggest that indeed the recently introducednew definitions of quantum non-Markovianity provide aweaker requirement with respect to the classical notion ofMarkovian classical process. In this respect, further andmore stringent notions of Markovian quantum dynamics cantherefore be introduced, e.g., relying on validity of the quantumregression theorem [17]. Our analysis however also shows thatthe non-Markovianity of the quantum dynamics, as assessedaccording to the trace distance criterion, appears to behavedifferently with respect to violation of the quantum regressiontheorem in the dependence of relevant model parameters suchas the coupling strength in the pure dephasing model. Despiteits simplicity, this model suggests that the features of beingMarkovian and of obeying the quantum regression theoremactually witness different aspects of the quantum dynamics.To better grasp this point further models should be consideredand analyzed in detail.

This fact further suggests critically discussing what shouldbe the meaning and relevance of a notion of non-Markovianquantum dynamics. Especially in view of the fact that a simpleand relevant hierarchy of non-Markovian quantum processes,analogous to the characterization of classical processes,appears unfeasible due to the intrinsic structure of quantummechanics, major emphasis should be put on the connectionbetween different indicators and observable properties. In thisrespect the quantum regression theorem refers to quantities,namely correlation functions, whose physical relevance is

well established; think, e.g., of the connection between powerspectra and autocorrelation functions. Among the newly in-troduced signatures of non-Markovian quantum dynamics, thenotion of Markovianity based on trace distance, without askingfor an explicit exact knowledge of the dynamical equations,allows for a direct experimental check, and has been mostrecently shown to be able to detect relevant modification in thesystem-environment dynamics, such as a phase transition [48].The analysis and comparison of utterly different quantities andconcepts which can be related to the notion of memory in aquantum dynamics, such as those considered in the presentpaper, will help in identifying their possible relevance inunveiling physical interesting phenomena.

ACKNOWLEDGMENTS

The authors gratefully acknowledge financial support by theEU projects COST Action MP 1006 and NANOQUESTFIT.

APPENDIX: ALTERNATIVE EXPRESSION OF THEDEPHASING FUNCTION

Starting from Eq. (40), namely

Ds(t) = �(s)

[1 + (�t)2]sin[s arctan(�t)], (A1)

and exploiting the identities

sin[arctan(x)] = x√1 + x2

, cos[arctan(x)] = 1√1 + x2

(A2)

together with

sin(sx) =∑k=0

(s

k

)[cos(x)]k[sin(x)]s−k sin

[π

2(s − k)

],

(A3)we can come to the compact expression (41)

Ds(t) = �(s)

2i[1 + (�t)2]s

[∑k=0

(s

k

)(�t)s−k[is−k − (−i)s−k]

]

= �(s)

2i[1 + (�t)2]s[(1 + i�t)s − (1 − i�t)s]

= �(s)Im[(1 + i�t)s]

[1 + (�t)2]s. (A4)

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